Math 360, Fall 2014, Assignment 8

Do not imagine that mathematics is hard and crabbed and repulsive to commmon sense. It is merely the etherealization of common sense.

- Lord Kelvin

Carefully define the following terms, then give one example and one non-example of each:

1. Direct product (of two groups).
2. Direct sum (of two abelian groups written additively).
3. Homomorphism.
4. Monomorphism.
5. Epimorphism.

Carefully state the following theorems (you need not prove them):

1. Theorem relating $\mathbb{Z}_m\times\mathbb{Z}_n$ to $\mathbb{Z}_{mn}$ (Theorem 11.5 in the text).
2. Fundamental Theorem of Finitely Generated Abelian Groups (Theorem 11.12).
3. Theorem concerning the value of a homomorphism at the identity.
4. Theorem concerning the value of a homomorphism on an inverse.

Solve the following problems:

1. Section 11, problems 1, 3, 8, 23, and 25.
2. Section 12, problems 1, 5, and 7.

--------------------End of assignment--------------------

Solutions:

Definitions:

1. Let $(G,\ast )$ and $(H, \vartriangle)$ be groups. Their direct product $G \times H$ is the set $G \times H$ under the operation $( g_1, h_1 )(g_2, h_2)=(g_1 \ast g_2 , h_1 \vartriangle h_2)$.
3. Let $(G, \ast)$ and $(H, \vartriangle)$ be groups. A homomorphism from $G$ to $H$ is a function $\phi : G \longrightarrow H$ that preserves $\phi ( g_1 \ast g_2) = \phi (g_1) \vartriangle \phi (g_2)$.
4. A monomorphism is a homomorphism that is injective.
5. An epimorphism is a homomorphism that is surjective.

Theorems:

1. The group $Z_m \times Z_n$ is cyclic and is isomorphic to $Z_{mn}$ if and only if $m$ and $n$ are relatively prime, that is, the gcd of $m$ and $n$ is 1.
2. Every finitely generated abelian group is isomorphic to a direct product of cyclic groups of the form \begin{equation*} \prod_{i=1}^{k}\mathbb{Z}_{p_i^{r_i}} \times \prod_{j=1}^n \mathbb{Z} \end{equation*} where the $p_i$ are primes (not necessarily unique) and the $r_i$ are positive integers. The factorization is unique except for a reordering of the factors, that is the $(p_i)^{r_i}$ are unique and the integer $n$ (the Betti number of $G$) is unique.
3. If $\phi$ is a homomorphism from $G$ to $H$, then $\phi(e_G)=e_H$.
4. If $\phi$ is a homomorphism from $G$ to $H$, then $\phi(a^{-1})=\phi(a)^{-1}$

Problems:

Section 11:

1. Elements:

• (0,0), order 1
• (0,1), order 4
• (0,2), order 2
• (0,3), order 4
• (1,0), order 2
• (1,1), order 4
• (1,2), order 2
• (1,3), order 4

There is no element of order 8, therefore the group is not cyclic. This can also be shown because the group is already in factorization as per Theorem 11.12, and since this factorization is unique, the group cannot be isomorphic to $\mathbb{Z}_n$ for any $n \in \mathbb{Z}$, and therefore cannot be cyclic.

3. The order of 2 in $\mathbb{Z}_4$ is 2 and the order of 6 in $\mathbb{Z}_{12}$ is 2, and $\operatorname{lcm}(2,2)=2$ so the order of $(2,6)$ in $\mathbb{Z}_4 \times \mathbb{Z}_{12}$ is 2.

8. The largest cyclic subgroup of $\mathbb{Z}_6 \times \mathbb{Z}_8$ is $\langle(2,1)\rangle$ with order 24. The largest cyclic subgroup of $\mathbb{Z}_{12} \times \mathbb{Z}_{15}$ is $\langle(3,1)\rangle$ with order 60.

23. Prime factorization of $32$ is $2^5$. Groups are:

• $\mathbb{Z}_{32}$
• $\mathbb{Z}_{16} \times \mathbb{Z}_{2}$
• $\mathbb{Z}_{8} \times \mathbb{Z}_{4}$
• $\mathbb{Z}_{8} \times \mathbb{Z}_{2} \times \mathbb{Z}_{2}$
• $\mathbb{Z}_{4} \times \mathbb{Z}_{4} \times \mathbb{Z}_{2}$
• $\mathbb{Z}_{4} \times \mathbb{Z}_{2} \times \mathbb{Z}_{2} \times \mathbb{Z}_{2}$
• $\mathbb{Z}_{2} \times \mathbb{Z}_{2} \times \mathbb{Z}_{2} \times \mathbb{Z}_{2} \times \mathbb{Z}_{2}$.

25. Prime facorization of $1089$ is $3^2*11^2$. Groups are:

• $\mathbb{Z}_{9} \times \mathbb{Z}_{121}$
• $\mathbb{Z}_{9} \times \mathbb{Z}_{11} \times \mathbb{Z}_{11}$
• $\mathbb{Z}_{3} \times \mathbb{Z}_{3} \times \mathbb{Z}_{121}$
• $\mathbb{Z}_{3} \times \mathbb{Z}_{3} \times \mathbb{Z}_{11} \times \mathbb{Z}_{11}$

Section 12:

1a. For a point $p \in \mathbb{R}$, the reflection through $p$ mapping $p+x$ to $p-x$ for any real $x$, and the identity map.
1b. The identity map, the rotations around the point in question, and the reflections along axes through the point in question.
1c. The identity map and the reflection through the midpoint of the line segment p, as in (a).
1d. The a rotation by $\pi$ around the midpoint of the line segment, a reflection through the midpoint along an axis perpendicular to the line segment, a reflection through an axis containing the line segment, and the identity mapping.
1e. Rotations of any angle about the axis containing the line segment, reflections across any plane containing the line segment, the reflection across the plane perpendicular to the line segment at its midpoint, and the identity mapping.